Sweep method for solving the Roesser type equation describing the motion in the pipeline

The initial problem for the system of hyperbolic equations describing the motion in oil production with gas lift method is considered. Introducing a new variable which is the difference of gas pressure and volume (or gas–liquid mixture (GLM)) multiplied by a constant number (balancing unit of measurements), the original system of equations is reduced to the such form of equations which after appropriate discretization becomes a Roesser type discrete equation. Searching the new variable as a linear function of the volume of gas (or GLM), it is shown that the coefficients satisfy the two difference equations of the first order, one of which corresponds to the quadratic equation and the second is a linear difference equation of the first order whose coefficients depend on the solution of the first one. In the case when the volume of the assessment gas and the motion (initial conditions) are constant at the mouth, it is shown that the results obtained by the Roesser model coincide with the known results, where the concrete analytical expression for the parameter of the balancing unit of measurements is provided.